Minimizing spatial-dispersion-induced birefringence

ABSTRACT

A composition formed from Group II fluorides in which the composition has little or no intrinsic birefringence at a selected wavelength. The composition is a mixed solid solution of CaF 2  with a second crystal of SrF 2  or BaF 2 . The resulting composition is in the form of Ca 1−x Sr x F 2  or Ca 1−x Ba x F 2 , or a combination of SrF 2  and BaF 2 , in the form of Ca 1−x−y Sr x Ba y F 2 . The specific form of the composition that effectively nulls out the intrinsic birefringence at a selected wavelength within the UV range is determined in one preferred method from the magnitudes of the intrinsic birefringences of the components, CaF 2 , SrF 2 , and BaF 2 .

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims benefit of the filing date of copendingProvisional Patent Application Nos. 60/303,898, filed on Jul. 9, 2001,and 60/309,192, filed on Aug. 1, 2001.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This invention was made by employees of the United States Government andmay be manufactured and used by or for the Government for governmentalpurposes without the payment of any royalties.

FIELD OF THE INVENTION

The present invention concerns birefringence in crystals used in opticalsystems, and more specifically to mixed solid solutions of fluoridecrystals in order to minimize spatial-dispersion-induced birefringencein components of optical systems.

BACKGROUND OF THE INVENTION

Fluoride crystals, such as calcium fluoride, strontium fluoride, andbarium fluoride, are widely utilized in high precision optics, includingUV optical lithography. These crystals exhibit high transmittance withina broad range of wavelengths from the infrared through the UV, below 157nm. Accordingly, these crystals are used in various kinds of opticalelements for precision UV optics, including lenses, prisms, and beamsplitters.

A major complication presently associated with crystals used inprecision optical systems is a phenomenon known as birefringence.Birefringence, also known as double refraction, refers to the dependenceof refractive index on light polarization direction. Most crystallinematerials are naturally birefringent and have anisotropic opticalproperties due to their asymmetric crystalline structure. However, it isgenerally thought that crystals with cubic-symmetry crystal structureare constrained by their high symmetry to have no inherent birefringenceand have isotropic optical properties. As a result, these cubic crystalswere believed to be ideal for use in precision optical systems.Birefringence in these crystals used in optical systems is generallythought to be primarily caused by mechanical stress or strainincorporated during the crystal fabrication process, and substantialefforts have gone towards reducing this stress-induced birefringence.

However, it has been recently discovered that cubic crystals such ascalcium fluoride, barium fluoride, and strontium fluoride have anintrinsic birefringence, in addition to the above-mentioned stressinduced birefringence as disclosed in John H. Burnett, Zachary H.Levine, and Eric L. Shirley, “Intrinsic Birefringence in 157 nmMaterials,” in R. Harbison, ed., 2 ^(nd) International Symposium on 157nm Lithography, (International SEMATECH, Austin, Tex., 2001); John H.Burnett, Zachary H. Levine, and Eric L. Shirley, “IntrinsicBirefringence in 157 nm Materials,” in R. Harbison, ed., CalciumFluoride Birefringence Workshop, (International SEMATECH, Austin, Tex.,2001); John H. Burnett, Zachary H. Levine, and Eric L. Shirley,“Intrinsic birefringence in calcium fluoride and barium fluoride,” Phys.Rev. B 64, 241102 (2001) (hereinafter “Intrinsic birefringence incalcium fluoride and barium fluoride”), all incorporated herein byreference. This birefringence is caused by the symmetry-breaking effectof the finite wave vector q of the photon, and is known asspatial-dispersion-induced birefringence, or intrinsic birefringence.This phenomenon was first discussed by H. A. Lorentz in 1878. It wasfirst convincingly demonstrated in 1971 in Si from J. Pastrnak and K.Vedam, in Phys. Rev. B 3, 2567 (1971) (hereinafter “Pastrnak”) and inGaAs from P.Y. Yu and M. Cardona, Solid State Commun. 9, 1421 (1971),both incorporated herein by reference, but the implications forprecision UV optics were not explored.

The problem of intrinsic birefringence must be addressed in precision UVoptical systems incorporating crystalline optics because the magnitudeof the birefringence in the UV is larger than the present industryspecifications, e.g., for 157 nm lithography as was reported by A. K.Bates, in Proceedings of the First International Symposium on 157 nmLithography, ed. by R. Harbison (International SEMATECH, Austin, 2000),p. 377, (hereafter “Bates”) incorporated herein by reference.

This complication presents serious challenges to optical engineersbecause, unlike stress-induced birefringence, intrinsic birefringence isinherent to the material, and thus cannot be reduced by materialimprovements in a single material.

Four main problems result from this intrinsic birefringence. The firstproblem is that a different refraction occurs for the two polarizationcomponents at the lens surface, which causes a ray bifurcation at eachlens. A second problem is that each polarization component accumulates adifferent phase as it transverses the crystal, resulting in phase-frontdistortion. A third problem is that an index anisotropy necessarilyaccompanies the birefringence. These combined effects of intrinsicbirefringence cause blurring of the image, which limits the achievableresolution. A fourth problem is an alteration of the polarization stateof light as it traverses the optics, which is significant for opticalsystems using polarized light. Accordingly, there exists a strong needto correct the problem of intrinsic birefringence in crystals used inhigh precision UV optical systems.

BRIEF SUMMARY OF THE INVENTION

The present invention concerns a method to eliminate or reduce theintrinsic birefringence in cubic crystals made from Group II fluorides,namely CaF₂, SrF₂, BaF₂, and MgF₂. The method is based on the discoverythat CaF₂ has a value of the intrinsic birefringence of opposite sign tothat of SrF₂ or BaF₂. As a result, mixed solid solutions of thesematerials, e.g., Ca_(1−x)Sr_(x)F₂, Ca_(1−x)Ba_(x), F₂,Ca_(1−x−y)Sr_(x)Ba_(y)F₂, Ca_(1−x−y)Sr_(x)Mg_(y)F₂, will have itsintrinsic birefringence nulled at a given wavelength with appropriatelychosen values of x (and y).

In accordance with one aspect of the present invention, a compositionincludes a mixture of CaF₂ crystal and a second crystal, saidcomposition having minimal spatial dispersion induced birefringence at aselected wavelength within the UV range.

In accordance with another aspect of the present invention, a method formaking a non-birefringent material includes selecting a wavelength, andmixing CaF₂ crystal with a second crystal to form a composition havingminimized spatial dispersion induced birefringence at the selectedwavelength.

In accordance with yet another aspect of the present invention, a deviceincludes an optical element formed from at least one compositioncomprising a mixture of CaF₂ crystal and at least one additionalcrystal. The composition is selected from the group consisting ofCa_(1−x−y)Sr_(x)Ba_(y)F₂, Ca_(1−x−y)Sr_(x)Mg_(y)F₂, andCa_(1−x−y)Ba_(x)Mg_(y)F₂, where x and y having values selected so as toform the composition with minimal intrinsic birefringence.

Further features and advantages of the present invention will be setforth in, or apparent from, the detailed description of preferredembodiments thereof which follows.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will now be described in detail with respect to preferredembodiments with reference to the accompanying drawings, wherein:

FIG. 1 is a plot depicting the relationship between intrinsicbirefringence and wavelength for CaF₂, SrF₂, and BaF₂ in accordance withthe present invention;

FIG. 2 is a plot depicting the intrinsic birefringence at variouswavelengths for compositions including those having different values ofSr fraction x; and

FIG. 3 is a plot depicting the relationship between intrinsicbirefringence and wavelength for compositions in accordance with thepresent invention including Ca_(1−x−y)Sr_(x)Ba_(y)F₂ where x is 0.3, yis 0.1.

DETAILED DESCRIPTION OF THE INVENTION

In accordance with one aspect of the present invention, there isprovided a composition formed from single-crystal mixed solid solutionsof Group II fluorides in which the composition has little or nointrinsic birefringence at a selected wavelength. The composition is amixed solid solution of CaF₂ with additional components of SrF₂, BaF₂,and/or MgF₂. CaF₂ has a negative sign for the intrinsic birefringencewhereas SrF₂ or BaF₂ has a positive sign of the intrinsic birefringence.Addition of Mg gives a negative contribution to the intrinsicbirefringence. The resulting composition is in the form ofCa_(1−x)Sr_(x)F₂ or Ca_(1−x)Ba_(x)F_(2,) or a combination of SrF₂, BaF₂,and MgF₂, in the form of, e.g., Ca_(1−x−y)Sr_(x)Ba_(y)F₂. The specificform of the composition that effectively nulls out the intrinsicbirefringence at a selected wavelength within the UV range is determinedfrom the magnitudes of the of the intrinsic birefringences of thecomponents, CaF_(2, SrF) ₂, and BaF₂. The effect of the addition of Mgis determined by calculated intrinsic birefringence of a theoreticalcubic-symmetry MgF₂, crystal.

In order to create the mixed crystal composition of the presentinvention, the intrinsic birefringences of the individual crystalcomponents are first determined by measurement and calculation. Themeasurement method is based on determining the amount of phasecompensation needed to null out light transmitted through an orientedsample between crossed polarizers in a way similar to that described inPastrnak, but with modifications for operation in the vacuumultraviolet, as described in “Intrinsic birefringence in calciumfluoride and barium fluoride.”

Measuring the effect as a function of wavelength in the VUV requiredoperation in an oxygen-free, nitrogen-purge environment with VUVpolarizers and a phase compensator, and using as illumination sources aseries of VUV atomic spectral lines. These were obtained from carbon Iemission from CO₂ introduced in a custom argon mini-arc lamp, filteredthough a monochromator, Bridges et al, Appl. Opt. 16, 367 (1977)incorporated herein by reference. The longer-wavelength UV measurementswere made using an electrodeless Hg lamp. The spectral line emission waschopped, collimated by a paraboloidal mirror, and linearly polarized bya MgF₂ Rochon polarizer. The polarized light passed through the sampleon a rotation stage and a MgF₂ Soleil-Babinet compensator oriented 45°to the polarizer, and impinged on a crossed MgF₂ Rochon polarizer. Lightthat was transmitted through the crossed polarizer, as a result ofbirefringence of the sample or compensator, was detected by a CsI orCsSb photomultiplier tube, using a lock-in amplifier. The amount ofphase compensation needed by the compensator to null out thebirefringence for a certain crystal orientation discussed below,determined the intrinsic birefringence of the material at a givenwavelength. The crystal geometry and orientation used for themeasurements were right rectangular parallelepipeds, with faces normalto the [110], [−110], and [001] crystallographic directions. The lightpropagation direction was in the [110] direction.

The value of the birefringence Δn=n_(<−110>)−n_(<001>) was determinedfor CaF₂, SrF₂, and BaF₂ for six wavelengths from 365.06 nm to 156.10nm. These were also estimated using an ab initio calculation, as wasdescribed in “Intrinsic birefringence in calcium fluoride and bariumfluoride.” The magnitudes and signs of the measurements and calculationsof Δn=n_(<−110>)−n_(<001>) for CaF₂, SrF₂, and BaF₂ near 193 nm and 156nm are shown in Table 1 and plotted in FIG. 1.

TABLE 1 Δn × 10⁷ Δn × 10⁷ Δn × 10⁷ Δn × 10⁷ (193 nm) (193 nm) (156 nm)(156 nm) MATERIAL (measured) (calculated) (measured) (calculated) CaF₂−3.4 ± 0.2  −1.3 −11.8 ± 0.4 −18 SrF₂ +6.6 ± 0.2  +9.8  +5.7 ± 0.3  +7.3BaF₂  +19 ± 2 +27   +33 ± 3 +52

These show that the magnitudes of the intrinsic birefringences are small(<10⁻⁷ for CaF₂) at long wavelengths but increase as the wavelength getsshorter (>10⁻⁶ for CaF₂ at 156.1 nm). The values of the birefringence at193.09 nm and at 156.10 nm are large compared to the low-birefringencerequirements of a number of precision UV high-numerical-aperture opticsapplications, in particular 193 nm lithography and 157 nm lithographytechnologies. The 157 nm lithography birefringence target specificationis Δn=1×10⁻⁷ for 157 nm as taught in Bates.

Both measurement and calculation shows that the sign of the effect forall materials are positive at long wavelengths, but the value for CaF₂has a zero crossing near ˜300 nm by measurement (or ˜200 nm by theory).For wavelengths below 200 nm, including 193 nm and 157 nm, the value forCaF₂ is negative, while that for SrF₂ and BaF₂ is positive. This is thekey result that enables a nulling of the intrinsic birefringence below200 nm for mixed solid solutions of CaF₂ with SrF₂ and/or BaF₂.

It is important that the birefringence be able to be nulled out for allpropagation directions in the crystal. That this is possible was firstdemonstrated by our symmetry analysis, summarize below, of the effectthat showed that the intrinsic birefringence is fully determined for agiven material at a given wavelength by a single parameter as describedin “intrinsic birefringence in calcium fluoride and barium fluoride.”

For a cubic crystal, the dielectric tensor may be expanded in wavevector, as shown in V. M. Agranovich and V. L. Ginzburg, Crystal Opticswith Spatial Dispersion and Excitons, 2^(nd) ed. (Springer-Verlag NewYork, 1984), pp. 129–135, which is incorporated herein by reference,ε_(ij)(q,ω)=ε(ω)δ_(ij)+Σ_(kl)α_(ijkl)(ω)q _(k) q _(l)+. . . ,  (1)where δ_(ij) is Kronecker's δ. The product q_(k)q_(l) and symmetry ofε_(ij) show that α_(ijkl) is symmetric under interchanges i⇄j and k⇄l.Further, cubic symmetry reduces the number of independent components tothe 3 familiar from elasticity theory of H. J. Juretschke, CrystalPhysics (Benjamin, London, 1974), Sections 4.2 and 11.2, which areincorporated herein by reference. An isotropic system permits twoindependent tensor components. One component represents aninconsequential change in ε_(ij)(q, ω) proportional to q²; the otherrepresents an isotropic longitudinal-transverse splitting. The thirdcomponent that exists in a cubic system (but does not exist in anisotropic system) determines all observable anisotropies. Only thiscomponent need be considered. The related tensor elements, α₁₁₁₁, α₁₁₂₂,and α₂₃₂₃, appear in the ratio 2:−1:−1, and this determines the angledependence. For {circumflex over (q)}=(1,1,0)/2^(1/2) the scaledeigenvalues of the contracted tensor Σ_(k)α_(ijkl)q_(k)q_(l) are 3/2 and−1 for the transverse [ 1 10] and [001] directions, respectively.Measuring the associated birefringence for one propagation directiondetermines the magnitude of the anisotropic response of the crystal forall propagation directions. The magnitude of the intrinsic birefringencein all directions is scaled by a single parameter. Nulling out thissingle parameter, eliminates the intrinsic birefringence for allpropagation directions.

The x value of the mixed solid solutions Ca_(1−x)Sr_(x)F₂ andCa_(1−x)Ba_(x)F₂ that nulls out the birefringence are predictedapproximately by Equation 2:i.x=|Δn(CaF ₂)/[Δn(CaF ₂)−Δn(YF ₂)]|,Y=Sr,Ba  (2)

The value obtained for Ca_(1−x)Sr_(x)F₂ and Ca_(1−x)Ba_(x)F₂ near 193 nmand 157 nm are shown in table 2.

As indicated above, these values assume a linear relation between thecomposition ratio and the birefringence parameter. Actually ourcalculations show that there is a small nonlinearity in thisrelationship, and thus a bowing in the composition/birefringence curves.Equation 1 is thus only approximately valid, and the exact nullingratios must be determined by measurement on the prepared crystals. FIG.2 shows calculations of the wavelength position of the birefringencenull of Ca_(1−x)Sr_(x)F₂ for different values of Sr fraction x.

TABLE 2 Nulls Intrinsic Nulls Intrinsic Birefringence at 193Birefringence at 156 Material nm nm Ca_(1-x)Sr_(x)F₂Ca_(0.66)Sr_(0.34)F₂ Ca_(0.34)Sr_(0.66)F₂ Ca_(1-x)Ba_(x)F₂Ca_(0.85)Ba_(0.15)F₂ Ca_(0.75)Ba_(0.25)F₂

It has previously been shown that such mixed solid solutions of CaF₂,SrF₂, and BaF₂ can be grown as a single crystal. For example E. G.Chernevskaya and G. V. Anan'eva, “Structure of Mixed Crystals Based onCaF₂, SrF₂, and BaF₂, ” Soviet Physics-Solid State 8, pp. 169–171 (1966)and separately R. K. Chang, Brad Lacina, and P. S. Pershan, “RamanScattering from Mixed Crystals (Ca_(x)Sr_(1−x))F₂ and(Ca_(x)Sr_(1−x))F₂, ” Phys. Rev. Lett. 17, pp. 755–758 showed thatCa_(1−x)Sr_(x)F₂ formed single crystal solutions with the fluorite(CaF₂) crystal structure for all ratios of components andCa_(1−x)Ba_(x)F₂ formed single crystal solutions with the fluorite(CaF₂) crystal structure at least when the composition of the componentsis low. The indices of CaF₂ and SrF₂ are fairly similar, e.g., n(CaF₂)near 157 nm=1.559 and n(SrF₂) near 157 nm=1 .576, a 1% difference as wasdemonstrated in John H. Burnett, Rajeev Gupta, and Ulf Griesmann,“Absolute refractive indices and thermal coefficients of CaF₂, SrF₂,BaF₂, and LiF near 157 nm,” Applied Optics 41, pp. 2508–2513 (2002).This means that the index of the mixture Ca_(1−x)Sr_(x)F₂ is relativelyinsensitive to unavoidable concentration gradients. Thus the materialcould have fairly uniform index, an important requirement for precisionoptics.

In addition to Ca, Ba, and Sr of the present mixed solid solutioncomposition, some fraction of Mg can be introduced in the mixed solidsolution to provide a composition having the form, e.g.,Ca_(1−x−y)Ba_(x)Mg_(y)F₂ or Ca_(1−x−y)Sr_(x)Mg_(y)F₂, without alteringthe cubic crystal structure. As shown in FIG. 2, Mg makes a negativecontribution to the intrinsic birefringence, as long as theconcentration is low enough that the material retains cubic structure.This increases the range of alloys available for intrinsic birefringencecompensation.

The introduction of Mg into the Ca_(1−x)Ba_(x)F₂ growth melt enableshigher solid solubilities of Ba in Ca_(1−x)Ba_(x)F₂. The higher solidsolubilities is provided by Mg having a smaller ionic radius compared tothat of CaF₂. Thus, Mg will compensate for the larger radius of BaF₂compared to that of CaF₂ to thereby enable higher solid solubilities ofBa in Ca_(1−x)Ba_(x)F₂.

By introducing quaternary alloys, e.g., Ca_(1−x−y)Sr_(x)Ba_(y)F₂ orCa_(1−x−y)Sr_(x)Mg_(y)F₂ or Ca_(1−x−y)Ba_(x)Mg_(y)F₂, a broadbandminimization of the intrinsic birefringence is possible. Appropriatevalues of x and y are chosen to minimize the intrinsic birefringencewithin the UV range such as, for example, the UV wavelengths of 193 nmand 157 nm. FIG. 3 shows calculations of the intrinsic birefringence ofBaF₂, SrF₂, CaF₂, and the quaternary mixed solid solutionCa_(0.6)Sr_(0.3)Ba_(0.1)F₂. The quaternary mixed solid solution shows abroadband minimization of the intrinsic birefringence, with an absolutemaximum value of 1.85 between 157 nm and 193 nm.

Further, the freedom to null out the birefringence for different x and yvalues enables the creation of intrinsic-birefringent-free materialswith different indices and dispersions. Optics with nulled birefringencefrom a family with distinct (x, y) values or for from different familiescan be combined for correction of chromatic aberrations (the minimizingthe derivative of the index of refraction with respect to sourcewavelength) due to the different index properties.

Although the invention has been described above in relation to preferredembodiments thereof, it will be understood by those skilled in the artthat variations and modifications can be effected in these preferredembodiments without departing from the scope and spirit of theinvention.

1. A composition comprising a mixture of CaF₂ crystal and a secondalkaline earth fluoride having spatial dispersion induced birefringenceopposite to the birefringence of the CaF₂ crystal the CaF₂ crystal andthe second crystal being present in amounts such that composition hasminimal spatial dispersion induced birefringence at a selectedwavelength within the UV range and said composition further comprisingMg.
 2. The composition of claim 1, wherein the composition is in theform Ca_(1−x−y)Ba_(x)Mg_(y)F₂.
 3. The composition of claim 1, whereinsaid second crystal is SrF₂.
 4. The composition of claim 3, wherein thecomposition is in the form Ca_(1−x−y)Sr_(x)Mg_(y)F₂.
 5. The compositionof claim 1, wherein said selected wavelength is between 193 to 157 nm.6. A composition comprising a mixture of CaF₂ crystal and a secondalkaline earth fluoride having spatial dispersion induced birefringenceopposite to the birefringence of the CaF₂ crystal the CaF₂ crystal andthe second crystal being present in amounts such that composition hasminimal spatial dispersion induced birefringence at a selectedwavelength within the UV range, said composition further comprising Sr,said second crystal comprising Ba, and said composition being in theform of Ca_(1−x−y)Sr_(x)Ba_(y)F₂.
 7. A method of making non-birefringentmaterial comprising the steps of: a) selecting a wavelength, and b)mixing CaF₂ crystal with a second alkaline earth fluoride having spatialdispersion induced birefringence opposite to the birefringence of theCaF₂ crystal and the CaF₂ crystal and the second crystal being presentin amounts such as to form a composition having minimized spatialdispersion induced birefringence at the selected wavelength, said secondcrystal comprising BaF₂.
 8. The method of claim 7, wherein thewavelength is selected within the UV range.
 9. The method of claim 7,wherein said mixing CaF₂ crystal with a second crystal comprises mixingCaF₂ with Ba to form the composition Ca_(1−x)Ba_(x)F₂ and selecting avalue for x to minimize the spatial dispersion induced birefringence atthe selected wavelength.
 10. A method of making non-birefringentmaterial comprising the steps of: a) selecting a wavelength, and b)mixing CaF₂ crystal with a second crystal to form a composition havingminimized spatial dispersion induced birefringence at the selectedwavelength, wherein the second crystal is SrF₂ and said mixing CaF₂crystal with a second crystal comprises mixing CaF₂ with SrF₂ to formthe composition Ca_(1−x)Sr_(x)F₂ and selecting a value for x to minimizethe spatial dispersion induced birefringence at the selected wavelength.11. The method of claim 7, wherein said mixing CaF₂ crystal with asecond crystal further comprises mixing CaF₂ with the BaF₂ and SrF₂ toform the composition Ca_(1−x−y)Sr_(x)Ba_(y)F₂ and selecting values for xand y to minimize the spatial dispersion induced birefringence at theselected wavelength.
 12. The method of claim 7, wherein said mixing CaF2crystal with a second crystal further comprises mixing CaF₂ with Ba andMg to form the composition Ca_(1−x−y)Ba_(x)Mg_(y)F₂ and selecting valuesfor x and y to minimize the spatial dispersion induced birefringence atthe selected wavelength.
 13. A method of making non-birefringentmaterial comprising the steps of: a) selecting a wavelength, and b)mixing CaF₂ crystal with a second crystal to form a composition havingminimized spatial dispersion induced birefringence at the selectedwavelength wherein the second crystal comprises SrF₂ and said mixingCaF₂ crystal with a second crystal further comprises mixing CaF₂ with Srand Mg to form the composition Ca_(1−x−Y)Sr_(x)Mg_(y)F₂ and selectingvalues for x and y to minimize the spatial dispersion inducedbirefringence at the selected wavelength.
 14. The method of claim 7,wherein said wavelength is between 157 to 193 nm.
 15. A devicecomprising: an optical element formed from at least one compositioncomprising a mixture of CaF₂ crystal and at least one additionalcrystal, said composition selected from the group consisting ofCa_(1−x−y)Sr_(x)Ba_(y)F₂, Ca_(1−x−y)Sr_(x)Mg_(y)F₂, andCa_(1−x−y)Ba_(x)Mg_(y)F₂, where x and y having values selected so as toform the composition with minimized intrinsic birefringence.
 16. Thedevice of claim 15, wherein the optical element comprises at least twocompositions selected from the group consisting ofCa_(1−x−y)Sr_(x)Ba_(y)F₂, Ca_(1−x−y)Sr_(x)Mg_(y)F₂, andCa_(1−x−y)Ba_(x)Mg_(y)F₂ with x and y have values to minimize oreliminate chromatic aberrations when electromagnetic energy interactswith said composition.
 17. A device comprising: an optical elementformed from at least one composition selected from the group consistingof Ca_(1−x−y)Sr_(x)Mg_(y)F₂, and Ca_(1−x−y)Ba_(x)Mg_(y)F₂, where x and yare values selected such that the composition has minimized intrinsicbirefringence.